[Edu-sig] Re: More on intro to Python (today's 3 hr training)

[Kirby Urner]

> Sorry for being picky, but while your derivative factory
> function follows the mathematical definition, it is not the
> "best" way to do it numerically.  The preferred way is:
> 
>   def derivative(f):
>       """
>       Factory function:  accepts a function, returns a closure
>       """
>       def df(x, h=1e-8):
>           return (f(x + h/2) - f(x - h/2))/h
>       return df
> 
> This is best seen by doing a graph (say a parabola) and drawing
> the derivative with a large "h" using both methods near a local
> minimum.
> 
> André

Hey good to know André and thanks for being picky.  

Fresh from yesterday's example, I posted something about *integration* to a
community college math teacher list last night, which I think is maybe
closer to your better way (which I hadn't seen yet).

===
 
def integrate(f,a,b,h=1e-3):
    """
    Definite integral with discrete h
    Accepts whatever function f, 
    runs x from a to b using increments h
    """
	x = a
	sum = 0	
	while x <= b:
	   sum += h*(f(x-h)+f(x+h))/2.0 # average f(x)*h
	   x += h
	return sum

  >>> def g(x):  return pow(x,2)  # so this is what we want to investigate

  >>> integrate(g,0,3,h=1e-4) # h = 0.0001, close to 9
  8.9995500350040558

  >>> integrate(f,0,3,h=1e-6) # h = 0.000001, even closer to 9
  8.9999954998916039

 def defintegral(intexp, a, b):
    return intexp(b) - intexp(a)

  >>> def intg(x):  return (1./3)*pow(x,3)  # limit function

  >>> defintegral(intg, 0, 3)  # exactly 9
  9.0

===

The post (which includes the above but is longer) hasn't showed up in the
Math Forum archives yet, or I'd add a link for the record.  Maybe later.

Kirby